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Logarithmic Components of the Vacant Set for Random Walk on a Discrete Torus

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Logarithmic Components of the Vacant Set for Random Walk on a Discrete Torus
2.	Creator	Author's name, affiliation, country	David Windisch; ETH Zurich
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	Giant component; vacant set; random walk; discrete torus
3.	Subject	Subject classification	60K35; 60G50; 82C41; 05C80
4.	Description	Abstract	This work continues the investigation, initiated in a recent work by Benjamini and Sznitman, of percolative properties of the set of points not visited by a random walk on the discrete torus $({\mathbb Z}/N{\mathbb Z})^d$ up to time $uN^d$ in high dimension $d$. If $u>0$ is chosen sufficiently small it has been shown that with overwhelming probability this vacant set contains a unique giant component containing segments of length $c_0 \log N$ for some constant $c_0 > 0$, and this component occupies a non-degenerate fraction of the total volume as $N$ tends to infinity. Within the same setup, we investigate here the complement of the giant component in the vacant set and show that some components consist of segments of logarithmic size. In particular, this shows that the choice of a sufficiently large constant $c_0>0$ is crucial in the definition of the giant component.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2008-05-09
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/506
10.	Identifier	Digital Object Identifier	10.1214/EJP.v13-506
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 13
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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